= Free straight edge of an isotropic bending plate
{title2=$w_{xx}+\nu w_{yy}=0,\quad w_{xxx}+(2-\nu)w_{xyy}=0$}
For a plate occupying $x>0$, the boundary part of the <first variation> of its <isotropic thin-plate bending energy> is, up to an overall nonzero sign,
$$
D\int_{x=0}[\eta_x(w_{xx}+\nu w_{yy})-
\eta(w_{xxx}+(2-\nu)w_{xyy})],dy.
$$
At a free edge the variation and its normal derivative can be prescribed independently. Their coefficients must therefore vanish, giving the two displayed natural boundary conditions: zero bending moment and zero effective transverse shear. The mixed-derivative coefficient includes the derivative along the edge of the twisting moment; keeping only the normal derivative of $w_{xx}+\nu w_{yy}$ would miss that contribution.
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